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7/23/2019 SHEAR STRENGTH OF SOIL (4).pptx
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EAG 345 –GEOTECHNICAL
ANALYSIS
By: Dr Mohd Ashraf MohamadIsmai
!iii" Mohr#Co$om% of S&r'ss
7/23/2019 SHEAR STRENGTH OF SOIL (4).pptx
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Mohr Circles & Failure Envelope
Y
σc
σc
σc
GL
As loading progresses,Mohr circle becomeslarger…
.. and fnally ailure occurswhen Mohr circle touchesthe envelope
∆σ
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Shear ailure mechanism
At ailure, shear stress along the ailure surace (τreaches the shear strength (τ .
σ τ
τ
σ τ
τ
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σΖ
σ!
τzx
Soil element
())
*+A
5) *+A
4) *+A
Mohr Circle of stress
Suppose a cubical sample o soil is sub"ected tothe stresses shown in the above fgure
#e would li$e to $now what the stresses at anypoint within the sample due to the applied
stresses.
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
% ($&a
'
($&a
)
*)
+))
+. he frst step is to plot a normal stress a!is(abscissa and shear stress a!is (ordinate
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
-. Assign a sign convention. #e will use anti*cloc$wise shear and compression as ve
% ($&a
'
($&a
( ), x zx
σ τ
( ), z zx
σ τ A !()),4)"
B !5),#
4)"
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
/. &lot the stresses. &oint A (+)),0)represents the stresses on the hori1ontalplane while point 2 (),*0) represents thestresses on the vertical plane. 3oin A2
A (+)),0)
2 (),*0)
% ($&a
'
($&a
( ), z zx
σ τ
( ), x zx
σ τ
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
0. #ith point ) as origin and 4A or 42 as radiusdraw a circle.
A (+)),0)
2 (),*0)
% ($&a
'
($&a4
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
. he ma"or principal stress is the value o thenormal stress at point +. he minor principalstress is the value o the normal stress atpoint /.
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
% ($&a
'
($&a
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
Alternatively5 the principal stresses are related to the stresscomponent '1 '! %1! by5
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
22
1
2
2
3
2 2
2 2
z x z x zx
z x z x zx
σ σ σ σ σ τ
σ σ σ σ σ τ
+ − = + + ÷
+ − = − + ÷
% ($&a
'($&a
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
Alternatively5
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
maxτ
1 3max
2
σ σ τ
−=
% ($&a
'
($&a
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
. #e will now determine the pole o Mohr6scircle. 7raw a line through A to represent thehori1ontal plane and a line through 2 torepresent the vertical plane. he intersectiono these planes is the pole, denoted by &.
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
% ($&a
'
($&a
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σΖ
σ!
τzx
())
*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+)
)
8. #e will now determine the pole o Mohr6s circle.7raw a line through A to represent thehori1ontal plane and a line through 2 torepresent the vertical plane. he intersection o
these planes is the pole, denoted by &.
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
&
% ($&a
'($&a
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&
Mohr Circle of stress
D C
BA
-
E
.
z σ
zxτ
zxτ
xσ
% ($&a
'($&a
A5 ( ), z xy
σ τ −
25 ( ), x xy
σ τ A ( ), z xy
σ τ −
2 ( ), x xy
σ τ
&oint A on the Mohr6s circle represents the stresses on the plane A2. Sothe line A& is drawn parallel to A2. &oint & become the &ole (& in thiscase
(&ole
9 we need to fnd stresses on a plane :;, we draw a line rom thepole parallel to :;, he point o intersection o this line with the
Mohr6s circle is < he coordinates o < give the stresses on the plane :;
/o' m'&hod of 01di12 s&r'ss's ao12 a
+a1'
.
<( ),n n
σ τ
1σ 3
σ
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σ6
2
'
3
'
1 σ σ +'
3σ '
1σ
PD = Pole w.r.t. plane
θ
(σ’, τ )
Orientation of Failure Plane
φ6
’1
’1
’3
’3
’
τ
’1
’1
’3
’3
’
τ
;ailure
envelope
(90 = θ)
hereore,
90 = θ + φ6 > θ
θ = 45 + φ62
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of stress
)
*)
+))
?. @et6s fnd the direction o the ma"or principalstress plane. A line is drawn rom the pole, & tothe point representing the principal stress, i.e…point +. the angle +&A is the inclination o thema"or principal plane to the hori1ontal and it ispositive as shown (anti*cloc$wise direction to
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
&
% ($&a
'
($&a
Horio1&a+a1'
Maor+ri1i+a
+a1'
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σΖ
σ!
τzx
())*/a
5) */a
4) */a
Mohr Circle of Stress
)
*)
+))
A (+)),0)
2 (),*0)
+/
3σ 1σ
&
3σ 1σ
% ($&a
'
($&a
ψ
Horio1&a+a1'
Maor +ri1i+a
+a1'
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σΖ
σ!
τzx
())*+A
5) *+A
4) *+A
Mohr Circle of Stress
)
*)
+)
)
A (+)),0)
2 (),*0)
+/
3 27.8σ = 1 122.2σ =
&
Alternatively5 the angle between the ma"or principal stress planeand the hori1ontal plane ( is5
1
tan zx
x
τ ψ
σ σ =
−
% ($&a
'($&a
2ψ
ψ
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σΖ
σ!
τzx
())
*/a
5) */a
4) */a
Mohr Circle of Stress
)
*)
+)
)
A (+)),0)
2 (),*0)
+/
3σ 1σ
&
B. o determine the stresses on any planeoriented at C rom the hori1ontal plane(cloc$wise rom the hori1ontal plane isnegative, draw a line rom the pole, & at an
angle C to the hori1ontal plane, A& here we
*C
θ τ
θ σ ( ),θ θ σ τ
% ($&a
'($&a
C
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Mohr Circle of Stress
)
*)
+)
)
A (+)),0)
2 (),*0)
+/
3σ 1σ
&*C
Alternatively5 the angle between the ma"or principal stress planeand the hori1ontal plane ( is5
1 3 1 3
1 3
co!22 2
!"n 2
2
θ
θ
σ σ σ σ σ θ
σ σ τ θ
+ −= +
−=
( ),θ θ σ τ
% ($&a
'($&a
σΖ
σ!
τzx
())
* /a
5) */a
4) */a
θ τ
θ σ
C
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Mohr Circle of Stress
)
*)
+)
)
A (+)),0)
2 (),*0)
+/
3σ 1σ
&*C
Alternatively5 the angle between the ma"or principal stress planeand the hori1ontal plane ( is5
co! 2 !"n 22 2
!"n 2 co! 2
2
z x z x
zx
z x
zx
θ
θ
σ σ σ σ σ θ τ θ
σ σ τ θ τ θ
+ −= + +
−= −
( ),θ θ σ τ
% ($&a
'($&a
σΖ
σ!
τzx
())
* /a
5) */a
4) */a
θ τ
θ σ
C
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Example 1
A sample o soil ().+ m ! ).+ m is sub"ected to the orces shown in
;ig. +. 7eterminea '+ ,'/ and
b he ma!imum shear stressc he stresses on a plane oriented at /)o countercloc$wise rom the
ma"or principal stress plane
σΖ
σ!
τzx
5 *N
3 *N
( *N
;ig. +
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Example 1
A sample o soil ().+ m ! ).+ m is sub"ected to the orces shown in
;ig. +. 7eterminea '+ ,'/ and
b he ma!imum shear stressc he stresses on a plane oriented at /)o countercloc$wise rom the
ma"or principal stress plane
σΖ
σ!
τzx
5 *N
3 *N
( *N
;ig. +
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Example 1
/ro'd$r':
here are two approaches to solve this problem. You can either useMohr6s circle or the appropriate eDuation. 2oth approaches will beused here.
(+ ;ind the area5 Area5
2#.1 #.1 #.#1 m A = × =
(- Ealculate the stress5
$orce 55## %Pa
area #.#1
33## %Pa
#.#11
1## %Pa#.#1
z
x
zx
σ
σ
τ
= = =
= =
= =
σΖ
σ!
τzx
))$&a
/))$&a
+))$&a
;ig. +
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Example 1
(/ 7raw Mohr6s circle and e!tract '+ ,'/ and %ma!
A()),+))
2 (/)),*+))
%
'
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
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Example 1
(/ 7raw Mohr6s circle and e!tract '+ ,'/ and %ma!
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
1
3
max
54# &Pa
2# &Pa
14# &Pa
σ
σ
τ
==
=
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
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Example 1
(0 7raw the pole on Mohr6s circle. he pole o Mohr6s circle is shown
by point & in the ;ig. below
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
/
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Example 1
A sample o soil ().+ m ! ).+ m is sub"ected to the orces shown in
;ig. +. 7eterminea '+ ,'/ and
b he ma!imum shear stressc he stresses on a plane oriented at /)o countercloc$wise rom the
ma"or principal stress plane
σΖ
σ!
τzx
5 *N
3 *N
( *N
;ig. +
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Example 1
( 7etermine . 7raw line rom & to '+ and measure the angle
between the hori1ontal plane and this line.
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
#22.5ψ =
Horio1&a/a1'
Maor +ri1i+as&r'ss +a1'
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Example 1
( Alternatively the angle between A4E can be used to determine
the
A ()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
C
##45
22.52 2
θ ψ = = =
CO
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Example 1
(8 7etermine the stresses on a plane inclined at /)o rom the ma"orprincipal stress plane. 7raw a line M6 and F6 through & with an
inclination o /)o rom the ma"or principal stress plane, angle E&F. he coordinate at point F is (0?), +-)
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
+))
-)
)
/))
*+))
*-))
*/))
+)
)
-)
)
/)
)
0)
)
)
)
8)
)
/)o
C
C
/N6
Coordi1a&' a& +oi1& N is!47),(8)"
M6 /a1' i1i1'd a&3)) from &h'
maor +ri1i+as&r'ss +a1'
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Example 1 (Alternative solution)
(/ 7raw Mohr6s circle and e!tract '+ ,'/ and %ma!
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
1
3
max
54# &Pa
2# &Pa
14# &Pa
σ
σ
τ
==
=
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
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Example 1 (Alternative solution)
Alternatively, we can use analytical solution rom the describedeDuation e!plained beore5
A()),+))
2 (/)),*
+))
3σ 1σ
maxτ
%
'
1
3
max
54# &Pa
2# &Pa
14# &Pa
σ
σ
τ
=
=
=
+))
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
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Example 1 (Alternative solution)
Analytical
solution5
2
2
1
22
3
2 2
2 2
z x z x zx
z x z x zx
σ σ σ σ σ τ
σ σ σ σ σ τ
+ − = + + ÷
+ − = − + ÷
1 3
max
2
σ σ τ
−=
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7/23/2019 SHEAR STRENGTH OF SOIL (4).pptx
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Example 1 (Alternative solution)
Ehec$
:Duilibrium5
8895o
5)) */a
()) */a
3)) */a
()) */a54(94*/a
!" Y
!";
(
83
@ength o -*/ > ).+ m
@ength o /*+ > ).+ ! (tan --.o >).)0+0 m
@ength o +*- > ).+ H (cos --.o >
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Example 1 (Alternative solution)Ehec$:Duilibrium5
8895o
5)) */a
()) */a
3)) */a
()) */a54(94*/a
!" Y
!";
(
83
#
#
# ( 3## #.#414 1## #.1 541.4 #.1#82 co!)22.5 * #
# ( 1## #.#414 5## #.1 541.4 #.1#82 !"n)22.5 * #
x
y
F
F
= − × − × + × × =
= − × − × + × × =∑∑
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Example 1 (Alternative solution)
Alternatively, we can use analytical solution rom the describedeDuation e!plained beore5
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
+)
)
-))
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
#
22.5ψ =
Horio1&a/a1'
Maor +ri1i+a
s&r'ss +a1'
1
1##tan #.414 22.5
541.4 3##
o zx
x
τ ψ
σ σ
= = = =
− −
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Example 1 (Alternative solution)
A()),+))
2 (/)),*+))
3σ 1σ
maxτ
%
'
+))
-)
)
/))
*+))
*-))
*/))
+))
-))
/))
0))
))
8))
/)o
C
C
/N6
Coordi1a&' a& +oi1& N is!47),(8)"
M6 /a1' i1i1'd a&3)) from &h'
maor +ri1i+as&r'ss +a1'
Alternatively, we can use analytical solution rom the describedeDuation e!plained beore5
( )
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Example 1 (Alternative solution)
Alternatively, we can use analytical solution rom the describedeDuation e!plained beore5
1 3 1 3
1 3
co!22 2
!"n22
θ
θ
σ σ σ σ σ θ
σ σ τ θ
+ −= +
−=
( )
( )
541.4 258. 541.4 258.co! 2 3# 47#.7 &Pa
2 2541.4 258.
!"n 2 3# 122.5 &Pa2
θ
θ
σ
τ
+ −= + =
−= =
H !
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Homeor!
()35*N<m8
=8(*N<m8
4(4*N<m8
4(4*N<m8
;or the stressed soil element shown in ;ig. abovedetermine5
a Ma"or principal stressb Minor principal stressc Formal and shear stresses on the plane A:
2y using graphical and analytical solution
Ieer to5 &rinciples o Geotechnical :ngineering (2ra"a, M. 7as